Questions on the circle, a curve composed of points in a plane that are at a fixed distance from a fixed point.

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2answers
362 views

Diameters and Circles

I have a question (given by a teacher) that looks really easy but then when I thought about it, couldn't find a way to find the answer. It is a proof question relating to diameters: Prove that any ...
23
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5answers
1k views

Did Euclid prove that $\pi$ is constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to ...
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0answers
73 views

Apollonius Circle help

I have been given a question which is very similar to this: Apollonius circle, its radius and center However, I have been told to translate and scale the given circle to give a unit circle centred ...
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1answer
185 views

Inversion applied on circles

I'm studying for my exam and one of the questions I am stuck on is: Show that under inversion in the unit circle a circle with centre C and radius $S$ inverts into a circle with centre ...
3
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3answers
169 views

Simple circle geometry/ similarity question

How would you prove that $a=b$ ? Would it be possible to solve this using similarity or trigonometry? Thank you in advance for any help. Any theorems or links would be appreciated.
2
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1answer
279 views

How to cut circle into $n$ parts (all cuts are parallel to each other) so that each chunk is the same area (i.e. $\pi r^2/n$)?

I have been working this problem for a few hours today, but I'm stuck. I started working on a case where $n = 3$: Let the radius of the circle be centered at $(0,0)$, with a radius $r$. The equation ...
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2answers
127 views

Circle-circle intersection on a spheroid

Does anybody have formulae to solve the following issue. If you have two circles, defined by their two centres, and a radius for each circle. Where (if the circles intersect) are the two points ...
0
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1answer
38 views

What are the applications? [duplicate]

How can I show that a sequence of regular polygons with n sides becomes more and more like a circle as n→∞? In which fields this concept is applied?
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1answer
76 views

what are the various fields in which circle is treated as infinite sided regular polygon?

What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided ...
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2answers
1k views

Circles in Complex Planes

Points on the circle centre C and radius r are given by the equation $|Z-C|=r$ or $(Z-C)(\overline{Z}-\overline{C})=r^2$. Where $Z = x + iy$. When multiplied out, I understand that we have ...
2
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3answers
440 views

A triangle has side lengths 4,6,8. A tangent is drawn to incircle parallel to side 4 cutting …

Problem : A triangle has side lengths $4,6,8$. A tangent is drawn to incircle parallel to side $4$ cutting other two sides at M and N, than length of MN is (a) $\frac{10}{9}$ (b) $\frac{20}{ 9}$ ...
2
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3answers
345 views

Geometry question involving a circle

If $P$ is a point inside a circle, how do you find the shortest distance from $P$ to the circumference of the circle?
2
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2answers
740 views

Circle Theorem - Alternate Segment Question

Hi there I have a maths question from my GCSE book which is just really bewildering me and my teacher. I have taken the maths question out from my book and made a computerized version and this is what ...
6
votes
0answers
229 views

What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two ...
4
votes
2answers
179 views

How is the circle that fits beneath two adjacent circles related? [duplicate]

This is hard to search and probably easy to solve, but I keep finding articles about intersecting circles, and that is not what I'm after. I don't know what to tag this under, so if you know how to ...
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0answers
106 views

Optimization and derivatives homework

Find the dimensions of a right circular cylindrical can with both a top and a bottom that holds 8 cubic cm and is constructed with the least amount of material possible. Radius of can= cm Height ...
2
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1answer
32 views

Approximate sector between two lines?

I need to approximate a red figure. I know coordinates of three points (little transparent circles). I also know a count of segments I need to divide this figure. The angle may be from 0 to Pi and ...
3
votes
1answer
128 views

Triangles within square

Points E and F lie on the sides BC and CD of rectangle ABCD, the AEF is an equilateral triangle. point M is the midpoint of the AF. Prove that the triangle BCM is equilateral.
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3answers
859 views

Find the radius and centre of the circle $x^2 -6x +y^2 -2y -6=0$

Find the radius and centre of the circle $x^2 -6x +y^2 -2y -6=0$ Can someone please help me with this question? I'm quite lost with what I have to do.
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1answer
795 views

find the equation of the circle passing through the extremities of the diameter of the circle

find the equation of the circle passing through the extremities of the diameter of the circle $x^2 +y^2 +2x-4y-2=0$ $x^2 +y^2 =0$ $x^2 +y^2 -6x-8y-2=0$ I cant understand what the question asks ...
0
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1answer
149 views

find the equation to the circle circumscribing the quadrilateral formed by the straight lines

find the equation to the circle circumscribing the quadrilateral formed by the straight lines $$2x+3y=2$$ $$3x-2y=4$$ $$x+2y=3$$ $$2x-y=3$$ we can see that the first two and the last two are ...
3
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2answers
1k views

Prove using integration that $polygon → circle\space \text{as}\space number\space of\space sides → infinity$

Say we have a regular polygon $s$, with number of sides $n$: Is there a way to prove that as $n → ∞,\space $then $s → circle$ using integration?
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2answers
44 views

Find the equation of a circle.

If a circle which center is on the straight line $x-2y+3=0$ cuts both the x-axis and the y-axis, what is the equation of the circle? ANSWER: $x^2+y^2-6x-6y+9=0$ This is rather a straightforward ...
4
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2answers
126 views

Triangle and its circumcircle

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. On sides $AC$, $BC$ lies respectively points $E$, $F$ such that $CE=BE$ and $CF=AF$. $CM$ is a median of triangle $EFC$. Show that line $CM$ pass ...
0
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1answer
599 views

Finding the condition for a straight line to be a tangent to a circle?

This is the question in my textbook-- Find the condition that the straight line $cx - by +b^2 = 0$ may touch the circle $x^2 + y^2 = ax + by $? My approach:- I made the distance of the center of ...
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1answer
51 views

Question about property of circle

We know that equal chords are equidistant from the center. However, I was curious if the lengths involved are proportional as well since the circle is a pretty symmetrical shape. Here's what I mean: ...
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5answers
360 views

Help in question related to locus of pair of tangent to a circle?

This the question in my text-book The tangent to $x^2 + y^2 = a^2$ having inclination $\alpha$ and $\beta$ intersect at $P$. If $cot\alpha$ + $cot\beta = 0$, then the locus of $P$ is : i really ...
8
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1answer
169 views

Is this curve the circumference of a circle?

Let $\Gamma$ be a single closed curve with no self-intersections on a plane which satisfies the following condition : Condition : For any distinct four points $P, Q, R, S$ on $\Gamma$, if the line ...
2
votes
1answer
105 views

Getting an angle

I have a unit circle, and two angles: $\alpha=\angle{JON}\in[0,\pi]$ and $\beta=\angle{IOM}\in[0,\frac{\pi}{2}]$. Using angles, we can get points $N$, $M$ as on the image. Then, dropping a ...
0
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0answers
43 views

Overlapping Circles Area [duplicate]

I have searched but could not find the exact question. Two circles with radii 5 intersect such that the center of one circle lies on the circumference of another. What is the area of the overlapping ...
1
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1answer
442 views

Finding the angle between 2 points on a circle

forgive me if this isn't the right place to ask this question but I am trying to figure out the value of theta along a line tangent to a circle from a starting position on the circle to an ending one ...
0
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1answer
57 views

Find a point given another point and angle

I have something starting at (50, 10) it then rotates counter clockwise by 30 degrees, around the point at (50, 0), essentially mapping out an arc of a circle. How do I find the point it now lies on?
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1answer
646 views

If inside a big circle , exactly n $(n \geq 3)$ small circles, each of radius r,can be drawn in such a way that each small circle touches t…

Problem : If inside a big circle , exactly n $(n \geq 3)$ small circles, each of radius r,can be drawn in such a way that each small circle touches the big circle and also touches its adjacent small ...
3
votes
1answer
97 views

Let P be a moving point such that if $PA$ and $PB$ are two tangents drawn from $P$ to the circle $x^2+y^2=1 ( $ A ,B being the points of contact) ,…

Problem : Let $P$ be a moving point such that if $PA$ and $PB$ are two tangents drawn from $P$ to the circle $x^2+y^2=1$ ( $A$, $B$ being the points of contact) , then $\angle AOB = 60^{\circ}$, ...
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3answers
897 views

Equation of a Circle from parametric functions of sin and cos

Given: x = 2 cos (t/2) y = 2 sin (t/2) How do we find the equation of the circle? I know that x^2 + y^2 = 1, where x = cos(t) y = sin(t) so x^2 = (2 cos (t/2))^2 y^2 = (2 sin (t/2))^2 How do ...
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1answer
55 views

Find the center of the circle… [closed]

Geometry | Circles | Equations of a Circle This is the equation for a circle: $$x^2 + y^2 + 4x + 2y - 11 = 0$$ Fill in the blanks for this answer box. Center: ( __ ,__ ) $$-$$ Radius: __
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1answer
340 views

Determine angle of incidence/reflection off of slanted line?

I'm working on an air hockey game (for learning purposes) and I'm currently struggling with some geometry. I'm trying to determine the new slope of the velocity of the puck when it collides with one ...
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2answers
872 views

Find the equation of a tangent line at $(3,-1)$ on the circle $x^2+y^2+2x-y-17=o$ [closed]

Determine the equations of tangents from point $A( 3 , -1)$ to circle (C) of equation: $x^2+y^2+4x+8y+3=0$ Thanks in advance :)
1
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1answer
716 views

Intersection points of two circles.

I understand that this is a common question and typically I can solve them, but this one keeps messing me up: Find the points of intersection (A and B) on the circles $x^2+y^2+4x-10y+20=0$ and ...
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1answer
84 views

Finding Area and probability[ hard nut to crack].

Suppose that $X$ and $Y$ are iid uniform distribution with $U(0, 1)$ random variables. (a) What is $\mathbb P((X, Y ) ∈ [a, b]×[c, d])$ for $0 ≤ a ≤ b ≤ 1$ and $0 ≤ c ≤ d ≤ 1$ ? What is $\mathbb ...
4
votes
1answer
142 views

What does relative height to the hypothenuse means?

I have to solve the next problem: Given H (relative height to the hypotenuse) and R (radius of the circle inscribed in the triangle) of a rectangle triangle, can you calculate the value of its ...
0
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4answers
214 views

How do I find the center and radius of this circle? [closed]

How do I find the center and radius of this circle? $$4x^2+4y^2+24x-16y+41=0$$
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3answers
247 views

Define “y” value from the equation of circle

Let's take a circle. It has the following general equation to describe it: $(x-u)^2+(y-v)^2=r^2$ ,where $u,v$ is the coordinates of the center of the circle, and $r$ is the radius of the circle. If ...
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1answer
916 views

How do I find the equation of the line that passes through this circle?

Find the equation of the line that has x-intercept and passes through the center of the circle that has equation $$x^2 + y^2-4x+10y+26=0$$
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1answer
235 views

Inscribed Angles/ Central Angles

If $\angle ACB = 40^{\circ}$(see figure), and the area of the circle is $81\pi$, how long is the arc $ABC$? This is how i have approached this problem. First of all we know the area of cricle is ...
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3answers
259 views

How to determine the number of degrees overlap between two circular slices?

How do you determine the number of degrees that overlap between two circular slices like what is shown in the example below by the hatched area? EDIT: Note, the slices are orientated by a center ...
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1answer
3k views

How many circles of radius r fit in a single bigger circle of radius R?

Is there any formula to calculate how many circles of radius r fit in a single bigger circle of radius R? I'd apreciate if it didn't involve advanced math, like calculus (unless there is no other way, ...
0
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3answers
154 views

Finding the equation of a circle ?

My Approach: I know that the general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c=0$. So, the aim is to fond the constants g,f,c.So, I should make equations relating these constants from the ...
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1answer
196 views

Finding all points around a circumference of a circle

I'm trying to write a program that lets the user put in the center point of a circle and its radius, and the put in two points to form a rectangle. Then I'm wanting it to print out whether the if the ...
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2answers
141 views

Preimage of a function

The only way to get better at this sort of thing is to practice, and now I'm also trying to ask myself (and try to answer) more conceptual questions. If a circle with radius $r$ is given in ...